physics help needed right now thanks?

Question

1. an astronaut lands on a planet that has twice the diameter and mass of earth. how does the astronaut weight differ from that of earth.

2.if the earth were uniform density [same mass and volumes throughout ] what would the value of g be inside the earth at half its radius.

3.the mass of a certain neutron stae is 3.0 x 10^30 kg [1.5 solar masses] and its radius is 8000 m [8 km]. what is the acceleratiin of gravity at the surface of this burnt out condensed star

Answer 1

All these can be resolved using F = GMm/r^2 (Newton's Law of Gravitation).

In the first case, weight will be: W = G * (2*Mass of Earth) * mass astronaut divided by (square of 2 *radius of earth).
Weight is found to be halved.

In the second case, we take the mass of the the sphere which is below the halfway point to the centre of the earth.

mass sphere = density * 4/3 * pi * (radius/2)^3
= (mass earth)/8

here g = G * (mass earth/8)/(radius/2)^2
= half its value at the surface of the earth.

for the third just apply the formula (acceleration due to gravity, g = GM/r^2) and obtain your value.
I obtained 12506250000000 m/s^2.

Answer 2

1. g=G*m/r^2 (g being surface gravitational acceleration, m being planet mass), so double r and m and you cut g in half.
2. The total g would be the sum of the g from the shell of matter outside the measurement point plus the g from the solid sphere inside that radius. From Newton's shell theorem discussed in the 2 references, the shell's g=0. (Ref. 1 is a detailed derivation, ref. 2 is brief and more intuitive.) Then you have the inside sphere's g, with half the radius and 1/8 the mass (.5^3=1/8). The result is g is half the surface value.
3. Just substitute m and r in g=G*m/r^2. (The value of G is not known to a high degree of precision. Typical values quoted are 6.672E-11 to 6.675E-11.)

Answer 3

1.Acceleration due to gravity of any planet is GM/R^2
as the planet has twice the diameter and twice the mass of the earth.
by substituting in the above formula we get the following result
astranaut will weigh half of that he will weigh on the earth


2.the value of g inside the earth at half of its radius is g/2
according to the formula g' = g(1-d/r) where d is the depth and r is the radius of the earth.

3.acceleration of gravity at the surface of this burnt out condensed star is 3.12*10^12
according to the formula g=Gm/r^2 where G is universal gravitational constant and its value is 6.67*10^11.